Possible? Yes. Practical? No.
As much as I prefer to do work by email, ultimately I do need to talk to my clients and vendors, perhaps in person. Skype does wonders in that I can often avoid travel and still "meet" in person, but to have no verbal conversation is nearly impossible.
The reason is that when things are going smoothly, email will suffice, but if an issue arises requiring clarification, insistence on using email alone can often compound the problem. In preparation for that, you talk on the phone at least (and preferably meet in person) so that should an issue arise, you've already set yourself up for better communication.
Additionally, email feels less personal than a phone call. Customers generally like knowing that there's a real person on the other side of the deal, and not that they are dealing with a faceless company. Having phone calls helps ground the personal relationship with the customers that makes them value working with you in particular.